First question, what instrument is that? Is it a trombone? Some type of a horn?

Second, why are they holding it like that? All three of them are doing it throughout the whole performance. Seems a little strange. Is it a stylistic thing? Is that how people held that instrument four hundred years ago?

1 Answer
1

It's a baroque trumpet, basically a historical version of a trumpet without valves. Probably they hold it like that because that was the way it was held at that time (think of musicians on a tower, announcing the arrival of the king or stuff like that...)

More specifically, baroque trumpets in D.
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American LukeAug 12 '13 at 13:43

The authentic Baroque trumpet, or natural horn, has no valves and no vents. Therefore it is possible to play it with one hand. It can only play the pitches of the D major scale; because there are no valves or vents, it is not possible to use the instrument to play additional notes or keys outside of D major.
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Wheat WilliamsAug 14 '13 at 11:48

2

Also note that the Baroque trumpet is usually tuned to A = 415, which is a half-step lower than the modern A = 440. Thus it sounds to your ear like the trumpet is playing in the key of D-flat and not D-natural.
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Wheat WilliamsAug 14 '13 at 11:50